An algorithm for solving the principal ideal problem with subfields

. The principal ideal principal (PIP) is the problem of deciding whether a given ideal of a number field is principal and, if it is, of finding a generator. Solving the PIP applies to solving major computational tasks in number theory. It is also connected to the search of approximate short vectors in the so-called ideal lattices which is a crucial problem in cryptography. In this paper, we present a novel application of norm relations to utilize information from subfields to solve the PIP in fields of degree larger than 1800.